Algebra 1: Factoring Practice Flashcards

Questions and Answers

What is the factorization of $72x^7y^2 - 16x^2y^3$?

8y²(9x^7 - 2x²)

2x^2(36x^5y - 8y²)

8x²y²(9x⁵ - 2y) (correct)

4x³y(18x^4 - 4y²)

What is the factorization of $-9y^9 - 18y^9x^2$?

-9y⁹(1 + 2x²) (correct)

-9y²(1 + 2yx²)

-18y^9(x + 1)

-9y^8(1 + 2xy)

What is the factorization of $20a^4b^5 + 8a^5b - 4a^3b$?

2a²b(10a^2b^4 + 4a - 2)

4a³b(5ab⁴ + 2a - 1)

8a²b²(2.5ab³ + a - 0.5)

4a³b(5ab⁴ + 2a² - 1) (correct)

What is the factorization of $x^4 + 3x^2 - 7x$?

x(x³ + 3x - 7) Signup and view all the answers

What is the factorization of $x^3 + 3x^2 + 4x + 12$?

(x² + 4)(x + 3) Signup and view all the answers

What is the factorization of $25xy + 30x - 30y - 36$?

(5x - 6)(5y + 6) Signup and view all the answers

What does Prime mean in factoring context?

When you try grouping, the two sets of parentheses do not match. Signup and view all the answers

What is the factorization of $16x^3 + 8x^2 + 8x + 4$?

4(2x² + 1)(2x + 1) Signup and view all the answers

What is the factorization of $n^2 - 8n + 7$?

(n - 7)(n - 1) Signup and view all the answers

What is the factorization of $x^2 - 6x + 4$?

Prime Signup and view all the answers

What is the factorization of $n^2 - 2n - 35$?

(n + 5)(n - 7) Signup and view all the answers

What is the factorization of $3x^2 - 15x + 12$?

3(x - 1)(x - 4) Signup and view all the answers

What is the factorization of $6x^2 - 42x - 108$?

6(x - 9)(x + 2) Signup and view all the answers

What is the factorization of $5x^2 + 27x + 10$?

(5x + 2)(x + 5) Signup and view all the answers

What is the factorization of $5v^2 - 23v + 12$?

(5v - 3)(v - 4) Signup and view all the answers

What is the factorization of $6k^2 - 3k - 9$?

3(2k - 3)(k + 1) Signup and view all the answers

What is the factorization of $10n^2 - 26n + 12$?

2(5n - 3)(n - 2) Signup and view all the answers

What is the factorization of $25m^2 - 1$?

(5m + 1)(5m - 1) Signup and view all the answers

What is the factorization of $16r^2 + 8r + 1$?

(4r + 1)² Signup and view all the answers

What is the factorization of $9k^2 - 12k + 4$?

(3k - 2)² Signup and view all the answers

What does Prime mean in the context of polynomials?

All of the above Signup and view all the answers

What is the factorization of $4x² + 13x + 9$?

(x + 1)(4x + 9) Signup and view all the answers

What is the factorization of $3n^2 - 75$?

3(n + 5)(n - 5) Signup and view all the answers

What is the factorization of $100b^2 + 120b + 36$?

4(5b + 3)² Signup and view all the answers

What is the factorization of $18a² - 24a + 8$?

2(3a - 2)² Signup and view all the answers

What is the factorization of $2x³ + 7x² - 2x - 7$?

(x-1)(x+1)(2x+7) Signup and view all the answers

What is the factorization of $12x³ - 20x² - 3x + 5$?

(2x-1)(2x+1)(3x-5) Signup and view all the answers

Study Notes

Factoring Basics

Factoring involves rewriting expressions as the product of their factors.
The greatest common factor (GCF) is the highest factor shared among terms.

Types of Factorization

GCF Only: Some expressions can be factored by identifying a common factor among all terms.
Factor by Grouping: Typically used for polynomials that can be grouped into pairs with common factors.
Trinomials: Quadratic expressions can often be factored into two binomials, especially when the leading coefficient (LC) is 1.
Special Cases: Certain forms, such as perfect square trinomials and differences of squares, follow specific factoring patterns.

Examples of Factoring

GCF Examples:
- ( 8x²y²(9x⁵ - 2y) ) becomes ( 72x⁷y² - 16x²y³ ).
- ( -9y⁹(1 + 2x²) ) leads to ( -9y⁹ - 18y⁹x² ).
Trinomial Factoring:
- ( n² - 8n + 7 ) is factored using ( (n - 7)(n - 1) ).
- ( n² - 2n - 35 ) factors to ( (n + 5)(n - 7) ).
Special Cases:
- The expression ( (5m + 1)(5m - 1) ) applies the difference of squares to yield ( 25m² - 1 ).
- ( (4r + 1)² ) expands to ( 16r² + 8r + 1 ) (perfect square trinomial).

Identifying Prime Expressions

Certain expressions cannot be factored further, labeled as "Prime," like ( x² - 6x + 4 ) where no pairs of numbers multiply to 4 and add to -6.

Application of GCF and Special Cases

Problems often combine GCF with special cases: ( 3(n + 5)(n - 5) ) simplifies to ( 3n² - 75 ) combining the difference of squares with GCF.

Summary of Key Techniques

Recognize the structure of the polynomial to determine the best factoring method.
Different methods may apply to the same polynomial depending on its form, such as grouping, recognizing trinomials, or applying GCF first.
Practicing various factoring methods helps in identifying the quickest path to the solution for polynomials.

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Description

Test your skills in factoring expressions with these flashcards designed for Algebra 1. Each card presents a problem to factor completely, focusing on the Greatest Common Factor (GCF). Use these cards to solidify your understanding and improve your algebra skills.